Armature bone rotation values not changing when using inverse kinematics
So let's look at one driven bone:

Here, Spine01 is getting a Y quaternion driver based on the Z quaternion component of Core_IK.  I've rotated Core_IK a bit, and showed Spine01's transform in the sidebar, so that you can see that, yes, the driver is actually working-- that Y rotation field is non-zero, and since Core_IK's Z component is -0.082, it's doing what you told it to.
But unfortunately, doing what you told it to do, is doing nothing.  Why?  The short answer is that the Y component is the only component you're changing.  With quaternions, increasing the Y component alone is not the same thing as rotating in the local Y axis.
Quaternions are kind of weird.  Even though Blender lets you deal with their components individually, they're not really four separate numbers; quaternions are normalized, four-dimensional vectors.  If you want Spine01 to rotate, you need to also change at least the W component as well as the Y.  How much?  Well, (w^2 + Y^2)^0.5 = 1.  So, that's why they taught you algebra way back when, for this moment.
Other driven bones in the armature have similar issues.
What should you do?  You should change the rotation mode of the bones that you want driven to be Euler bones.  You can specify the rotation mode in properties/bone/transform/mode.  If you do that, your individual channels will represent rotation in a particular axis.  In the case of Spine01, which looks like you're only ever going to use drivers to control its roll, there's no advantage to quaternions over Eulers anyways.  The advantages to quaternion interpolation are only ever apparent when a bone rotates in more than one Euler axis.
Is there anything else you should change?  Yes.  Just as the Y quat component of Spine01 doesn't rotate anything, the Z quat component of Core_IK is not a very useful input to drive this rotation.  Here, you don't need to change the rotation mode of anything, you just need to change how Blender reads whatever transform you give it, by changing the mode in the &quot;rot&quot; variable for your driver.  You can leave Core_IK as a quaternion bone and still read its orientation as an Euler for purposes of rotating Spine01.